| L(s) = 1 | + 5-s + 2·7-s − 6·11-s − 13-s + 2·17-s + 2·19-s − 4·23-s + 25-s − 2·29-s + 2·31-s + 2·35-s + 2·37-s + 6·41-s + 6·47-s − 3·49-s + 2·53-s − 6·55-s − 6·59-s + 14·61-s − 65-s − 2·67-s − 10·71-s − 6·73-s − 12·77-s − 4·79-s + 2·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.80·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.338·35-s + 0.328·37-s + 0.937·41-s + 0.875·47-s − 3/7·49-s + 0.274·53-s − 0.809·55-s − 0.781·59-s + 1.79·61-s − 0.124·65-s − 0.244·67-s − 1.18·71-s − 0.702·73-s − 1.36·77-s − 0.450·79-s + 0.219·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043274015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043274015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63260264468231271767657011796, −7.33535606041816470770626790384, −6.12365705266944617109077027258, −5.63205115874468158579655890780, −5.00386328721117394492261016605, −4.40268793334971864544129664061, −3.30818001293029430401084271723, −2.51533084366846836447331869803, −1.87669537418863006009072377774, −0.67245244112990566449720195588,
0.67245244112990566449720195588, 1.87669537418863006009072377774, 2.51533084366846836447331869803, 3.30818001293029430401084271723, 4.40268793334971864544129664061, 5.00386328721117394492261016605, 5.63205115874468158579655890780, 6.12365705266944617109077027258, 7.33535606041816470770626790384, 7.63260264468231271767657011796
